in which \(N\)  is the number of lattice sites, \(x\)  denotes the
coordinates of lattice sites and \(n_x\)  is generally related to the
physical quantities located at site \(x\) , such as the total spin or
charge, or particle number at site \(x\)  and so on. The twist
operator was first introduced by Lieb, Schultz and Mattis for the
proof of gapless excitation in one-dimensional spin-1/2
chains[1]}. Then Resta pointed out that its ground state
expectation was direct to the Berry-phase theory of polarization in
strongly correlated electron systems [2]}. Moreover Aligia
found that the ground-state expectation value of the twist operator
Eq. (REF ) allows one to discriminate conducting from
nonconducting phase in the extended quantum systems [3]}.
The vanishing of the ground-state expectation value, i.e. \(\eta =0\) ,
has been shown the ability to detect the boundaries of different
valence-bond-solid phases in spin chains [4]}. However these
studies were implemented in some special examples and a general
discussion was absent. Moreover since the previous calculations were
numerical or approximate, the details of \(\eta \)  adjacent to the
phase transition points are unclear. Hence it is of great interest
to find the exact expression for \(\eta \) , even for special cases. Our
paper serves this purpose, and the exact results can be found for a
special case.
